Application of the Reference-Spectrum Method to the Nuclear Surface

The reference-spectrum method for infinite nuclear matter is generalized in a simple way so as to apply to the surface region of a large nucleus. This permits one to calculate the reaction matrix for particles interacting in the surface region, and the total energy of the nucleus can therefore be found to first order in the Brueckner-Goldstone expansion. Two equivalent formulas for the nuclear surface energy, both valid to first order in the Brueckner-Goldstone expansion, are derived and discussed. These results are then used to calculate the surface energy of a large nuclear slab. The single-particle wave functions are calculated in a Woods-Saxon potential which is chosen to reproduce the empirical surface thickness; and the two-body interaction, which acts only in $S$ states, is assumed to consist of an exponential attraction outside a hard core. The calculation gives the reasonable value of $20.6 A^{\frac{2}{3}}$ MeV for the surface term in the semiempirical mass formula. The chief defect of this trial calculation is the lack of self-consistency in the one-particle wave functions. A self-consistent calculation would give a more reliable estimate for the surface energy and would also predict the surface thickness theoretically. A brief discussion is given of how the methods of this paper could be used to attack the self-consistency problem.